Proposition 9.4.6.17. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. Then $U$ is exponentiable if and only if it is a flat isofibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Assume that $U$ is a flat isofibration; we will show that $U$ is exponentiable (the converse follows from Remark 4.5.9.12 and Example 9.4.6.2). Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}'' \ar [r]^-{ F} \ar [d] & \operatorname{\mathcal{E}}_0 \ar [r]^-{G} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}_1 \ar [r]^-{ \overline{F} } & \operatorname{\mathcal{C}}_0 \ar [r]^-{ \overline{G} } & \operatorname{\mathcal{C}}} \]
in which both squares are pullbacks, where $\overline{F}$ is a categorical equivalence; we wish to show that $F$ is also a categorical equivalence. Using Proposition 4.1.3.2, we can factor $\overline{G}$ as a composition $\operatorname{\mathcal{C}}_0 \xrightarrow {\iota _0} \operatorname{\mathcal{C}}_{0}^{+} \xrightarrow {V_0} \operatorname{\mathcal{C}}$, where $\iota _0$ is inner anodyne and $V_0$ is an inner fibration. In particular, $\operatorname{\mathcal{C}}^{+}_{0}$ is an $\infty $-category. Replacing $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}^{+}_{0}$ (and $U$ by the projection map $\operatorname{\mathcal{C}}_0^{+} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}^{+}_{0}$), we can assume that $\overline{G}$ is inner anodyne. In this case, Corollary 9.4.6.7 guarantees that $G$ is a categorical equivalence of simplicial sets. It will therefore suffice to show that the composition $(G \circ F): \operatorname{\mathcal{E}}_1 \rightarrow \operatorname{\mathcal{E}}$ is a categorical equivalence. Applying Proposition 4.1.3.2 again, we can factor $\overline{G} \circ \overline{F}$ as a composition $\operatorname{\mathcal{C}}_1 \xrightarrow {\iota _1} \operatorname{\mathcal{C}}_{1}^{+} \xrightarrow {V_{1}} \operatorname{\mathcal{C}}$, where $\iota _1$ is inner anodyne and $V_1$ is an inner fibration. Applying Corollary 9.4.6.7 again, we conclude that the inclusion map $\operatorname{\mathcal{E}}_1 \hookrightarrow \operatorname{\mathcal{C}}_{1}^{+} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a categorical equivalence of simplicial sets. We are therefore reduced to showing that the projection map $\operatorname{\mathcal{C}}_{1}^{+} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}$ is an equivalence of $\infty $-categories. Note that $\overline{F}$, $\overline{G}$, and $\iota _{1}$ are categorical equivalences of simplicial sets, so the inner fibration $V_1$ is an equivalence of $\infty $-categories. The desired result now follows from Corollary 4.5.2.29 (since $U$ is an isofibration). $\square$